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University of Colorado, Boulder
CU Scholar
Undergraduate Honors Theses
Honors Program
Spring 2017
Inferring Telescope Polarization Properties
Through Spectral Lines Without Linear
Polarization
Alysa Derks
[email protected]
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Recommended Citation
Derks, Alysa, "Inferring Telescope Polarization Properties Through Spectral Lines Without Linear Polarization" (2017). Undergraduate
Honors Theses. 1325.
http://scholar.colorado.edu/honr_theses/1325
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Inferring Telescope Polarization Properties Through Spectral Lines
Without Linear Polarization
Alysa Derks
Department of Astrophysical and Planetary Sciences
University of Colorado, Boulder
Thesis Advisor: Valentin M. Pillet, APS
Honors Council Representative: Erica Ellingson, APS
Other Committee Members: Kelvin Wagner, ECEN
Christian Beck, National Solar Observatory
Defended: April 10, 2017
1
1
Abstract
Polarimetric observations taken with ground- or space-based telescopes usually need to be corrected for changes of the polarization state induced in the optical path. In this paper, I present a
new technique to determine the polarization properties of a telescope through observations of spectral lines that have no or negligible intrinsic linear polarization 1 signals. For such spectral lines,
any observed linear polarization must be induced by the telescope optics. I apply the technique
to observations taken with the Spectropolarimeter for Infrared and Optical Regions (SPINOR) at
the Dunn Solar Telescope (DST) and demonstrate that I can retrieve the characteristic polarization properties of the DST at three selected wavelengths of 459, 526 and 615 nm. I determine
the amount of cross-talk between the intensity, Stokes I, and the linear and circular polarization
2 states, Stokes Q, U and V , and between Stokes V and Stokes Q and U in spectropolarimetric observations of active regions on the solar surface. I fit a set of parameters that describe the
polarization properties of the DST to the observed cross-talk values 3 . I compare my results to
parameters that were derived using a conventional telescope calibration unit (TCU). The values
for the ratio of reflectivities X = rs /rp 4 and the retardance τ of the DST turret mirrors from
the analysis of the cross-talk match those derived with the TCU within the error bars. I find a
negligible contribution of retardance from the entrance and exit windows of the evacuated part of
the DST. Residual cross-talk after applying a correction for the telescope polarization stays at a
level of 3-10 % regardless of which parameter set is used, but with an rms fluctuation in the input
data of already a few percent. The accuracy in the determination of the telescope properties is thus
more limited by the quality of the input data than the method itself. It is possible to derive the
parameters that describe the polarization properties of a telescope from observations of spectral
lines without intrinsic linear polarization signal. Such spectral lines have a dense coverage (about
every 50 nm) in the visible part of the spectrum (400–615 nm), but none were found at longer
wavelengths. Using spectral lines without intrinsic linear polarization is a promising tool for the
polarimetric calibration of current or future solar telescopes such as the Daniel K. Inouye Solar
Telescope (DKIST).
1
linear polarization = light in the form of a plane wave
circular polarization = light composed of two plane waves of equal amplitude, differing in phase by 90 deg.
3
polarization cross-talk is spurious polarization signal caused by transfer from one Stokes parameter into another
one by polarimetrically active optical elements
4
rs is the reflectivity in the plane perpendicular to the plane of incidence. rp is the reflectivity in the plane parallel
to the plane of incidence.
2
2
Contents
1 Abstract
2
2 Introduction
4
3 Synthesis of Spectral Lines and Line Selection
5
4 Application of Lines without LP for Polarimetric Calibration
9
5 Observations
10
6 Data Analysis
6.1 Preparation of Input Data . . . . . . . . . . . . . .
6.1.1 Reduction of SPINOR Data . . . . . . . . .
6.1.2 Determination of Locations with Significant
6.1.3 Determination of V → QU Cross-Talk . . .
6.2 Fit of DST Telescope Model . . . . . . . . . . . . .
6.2.1 DST telescope model . . . . . . . . . . . . .
6.2.2 Input Data for Fit of Telescope Model . . .
6.2.3 Fit of Telescope Model Parameters . . . .
7 Results
7.1 Fit Quality . . . . . . . . . . . . . . . . . .
7.2 Ratio of Reflectivities X = rs /rp . . . . . .
7.3 Window and Mirror Retardance τ . . . . .
7.4 Sensitivity Test and Error Estimate . . . .
7.5 Residual cross talk with telescope correction
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Polarization
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8 Summary & Discussion
25
8.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
8.2 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
9 Conclusions & Future Work
26
10 Acknowledgments
27
A Appendix A: Line spectra at 426, 431 and 514 nm
29
B Appendix B: Transition parameters and polarization amplitude
32
3
2
Introduction
Measuring the polarization state of light, or polarimetry, is important because we can use
these observations to better protect ourselves from solar flares. Incoming light from the sun is
unpolarized. Some telescope optics can change this, creating polarization and making observers see
a polarization that shouldn’t really be there. Therefore, this induced telescope polarization needs
to be determined and compensated for.
Most existing solar telescopes were designed without clear requirements about their polarimetric
performance and calibration. The telescopes currently available for high spatial resolution solar
physics are calibrated using an analytical model of their time-varying Mueller matrices 5 . The free
parameters in these models are fit to data obtained using calibration polarizers at the entrance of
the telescopes (Skumanich et al. (1997); Beck et al. (2005); Selbing (2010); Socas-Navarro et al.
(2011)) or they used literature values for the polarization properties of the mirrors (see Collados
(1999); Schlichenmaier & Collados (2002)).
The calibration strategy of the Advanced Stokes Polarimeter (ASP; Elmore et al. 1992) at the
Dunn Solar Telescope (DST Dunn 1964; Dunn & Smartt 1991) as described in Skumanich et al.
(1997) paved the way for other telescopes to carry out similar polarimetric characterizations. A
key aspect of the ASP/DST calibration was to split the polarization effects of the telescope with
a time-dependent Mueller matrix T(t) and those of the polarimeter with a static Mueller matrix
X by inserting polarization calibration optics between the two. This point of insertion made the
polarimeter matrix X constant in time but made the telescope Mueller matrix T time-dependent to
account for the varying telescope geometry. The calibration optics provided the polarization properties of all of the optical components downstream from the insertion point, i.e., the polarimeter
matrix. Upstream from the insertion point, the data obtained with the entrance polarizer constrained the polarization properties described by the matrix T(t). Polarization calibration optics
usually have a small diameter and can be manufactured to high accuracy compared to the large
optics used for the entrance polarizers. Placing such high-quality calibration optics as high up in
the optical path as feasible is thus beneficial.
All modern solar telescopes adopted the strategy of the ASP and incorporated calibration optics
as early in the optical path as possible, thus calibrating the largest number of optical components.
One recent example is the German GREGOR telescope (Schmidt et al. 2012) that uses a polarization
calibration unit near the secondary focus of the telescope, right after the primary and secondary
mirrors. The calibration unit is located before breaking the revolution symmetry around the optical
axis. Thus, the expected polarization effects from the mirrors upstream are small, originating only
from the off-axis configuration outside the center of the field of view, and from diffraction effects.
Sanchez Almeida & Martinez Pillet (1992) demonstrated that these effects are only important for
absolute polarimetry at the 10−4 level. For GREGOR, the matrix downstream of the calibration
optics is geometry and time-dependent, whereas the matrix upstream is constant in time and, in
this case, close to the identity matrix.
The Daniel K. Inouye Solar Telescope (DKIST; McMullin et al. 2016) under construction on the
island of Maui will follow a similar strategy as GREGOR. A polarization calibration unit is located
downstream of the secondary mirror. In the case of DKIST, the first two mirrors M1 and M2 (see
Fig. 1) have an off-axis configuration that generates a telescope matrix T that deviates significantly
from the identity matrix, and is constant in time. The angles of incidence of these two mirrors vary
from 12 to 14 degrees and non-diagonal matrix elements of a few 10−2 are expected. Accurate raytracing with realistic modeling of the mirror coatings demonstrates that Stokes U ↔ V cross-talk
5
a 4 × 4 matrix that holds the polarization properties of a telescope.
4
terms on the order of 8% can be expected in T (Harrington & Sueoka 2016, 2017). Interestingly,
the same calculations show that the diagonal elements are all close to unity to within less than a
percent. Similar to GREGOR, all time dependencies are transferred to the matrix downstream of
the calibration optics. The polarization properties of T are independent of the telescope pointing
and change only with the degradation of the coatings with time. Thus, the inference of this matrix
is much simpler than in the case of the DST.
The 4-m aperture of DKIST makes it impossible to use an entrance linear polarizer. A number
of strategies have been devised to calibrate the instrumental polarization to the required level of
accuracy of 5 × 10−4 (see Socas-Navarro et al. 2005; Socas-Navarro 2005a,b; Elmore et al. 2014).
One option for the calibration of the DKIST T-matrix is based on spectral lines that have no
intrinsic linear polarization through the Zeeman effect, e.g., the Fe ii line at 614.9 nm (Lites
1993). A list of lines with this property, which depends on the atomic transitions involved, was
compiled by Sanchez Almeida & Vela Villahoz (1993) and Vela Villahoz et al. (1994). These authors
proposed the use of those lines to measure the asymmetries of their Stokes V profiles as they are
unaffected by instrumental polarization. In this paper, I revisit their list of lines without linear
polarization to study their suitability for polarimetric calibration. I compare the results obtained
from observations of lines without linear polarization taken in 2016 at the DST to those from the
standard calibration done in 2010 using the telescope entrance linear polarizers to understand the
limitations of the approach. The calibration of the T(t) matrix for the DST is more complex than
the calibration of T for DKIST, thus succeeding with the DST demonstrates the potential interest
of this method for DKIST. This approach was first proposed for DKIST by Socas-Navarro et al.
(2005) who demonstrated that it can potentially achieve the required accuracy.
Section 3 describes my selection process to identify suited spectral lines. Section 4 gives the theoretical background for the derivation of telescope properties from observations. The observations
used are described in Section 5. Sections 6 and 7 provide the data analysis and results, respectively.
Section 8 contains the discussion, while Section 9 gives my conclusions. The appendices provide
the complete line list and a few more data examples at other wavelengths.
3
Synthesis of Spectral Lines and Line Selection
There exist special atomic transitions that produce spectral lines without any, or with negligible,
intrinsic linear polarization (LP) induced by the Zeeman effect (Sanchez Almeida & Vela Villahoz
(1993); Vela Villahoz et al. (1994)). These lines can be used to uncover polarization properties of
a telescope or to test the quality of the polarimetric calibration because most linear polarization
signal is spurious and indicates polarization cross-talk. I analyzed the two tables of spectral lines
without or with small LP provided by Vela Villahoz et al. (1994, their Tables 1 and 2) to find the
lines that would be best suited for telescope calibration purposes. I restricted the analysis to lines
with a rest wavelength from 400–1100 nm. I obtained all necessary information including excitation
potential, log(gf)6 , transition parameters, and the effective Landé coefficient (geff )7 from the NIST
Atomic Spectral Database8 . I disregarded any lines for which the value of log(gf) could not be
found. This left 31 lines from Table 1 of Vela Villahoz et al. (1994) (see Table 4) and 56 lines from
their Table 2 (see Table 5).
After the suitable lines were chosen, a spectral synthesis was done using the Stokes Inversion
6
the logarithm of the multiplicity of the level times the oscillator strength, respectively.
measures the reaction of an electron transition to magnetic fields.
8
http://physics.nist.gov/PhysRefData/ASD/lines form.html
7
5
Figure 1: A schematic of the DKIST telescope.
6
1.2
0.04
1.0
0.8
0.02
I/I_c
V/I_c
0.6
0.00
0.4
−0.02
0.2
0.0
−0.04
−0.2
−0.2
−0.1
0.0
0.1
0.2
−0.2
−0.1
Wavelength (nm)
0.0
0.1
0.2
Wavelength (nm)
Figure 2: Synthetic Stokes profiles of Stokes I (left panel) and V (right panel) for the Cr ii line at
458.820 nm in the synthesis with 1500 G are shown in red. The blue lines show the corresponding
profiles from the FTS atlas.
1.2
0.04
1.0
0.8
0.02
I/I_c
V/I_c
0.6
0.00
0.4
−0.02
0.2
0.0
−0.04
−0.2
−0.2
−0.1
0.0
0.1
0.2
−0.2
−0.1
Wavelength (nm)
0.0
0.1
0.2
Wavelength (nm)
Figure 3: Same as Fig. 2 for Fe ii at 614.923 nm.
based on Response functions code 9 (SIR; Ruiz Cobo & del Toro Iniesta 1992). For this synthesis,
I used a solar model atmosphere with a magnetic field inclination of 45 ◦ and an azimuth of 22.5 ◦ ,
a macro- and micro-turbulent velocity of 1 km s−1 , and zero Doppler shift. The magnetic field
inclination and azimuth were chosen in order to obtain polarization signal in all Stokes parameters
Q, U , and V (without these specifications, only I and V signals would have been produced), whereas
the macro- and micro-turbulence were chosen so that realistic polarization amplitudes would be
obtained. The synthesis was repeated with three different magnetic field strengths: 100 G (quiet
Sun), 1500 G (plage), and 2500 G (sunspot) to simulate regions of different magnetic activity on
the solar surface.
Using the synthetic profiles, I made a final selection of lines suitable for calibration purposes
based on line depth, polarization amplitude, and absence of line blends. Line blends were identified
by a comparison to spectra from the Fourier Transform Spectrometer Atlas (FTS; Kurucz et al.
1984). I extracted the line depths from the 100 G synthesis, and the maximal Stokes Q, U , and
9
SIR synthesizes spectral lines formed in the presence of magnetic fields. It calculates the Stokes Spectra emerging
from any specified model atmosphere by numerically solving the radiative transfer equation for polarized light.
7
1.2
0.020
0.015
1.0
0.010
0.8
0.005
I/I_c
Q/I_c
0.6
0.000
0.4
−0.005
0.2
−0.010
0.0
−0.015
−0.2
−0.020
−0.2
−0.1
0.0
0.1
0.2
−0.2
−0.1
Wavelength (nm)
0.0
0.1
0.2
0.1
0.2
Wavelength (nm)
0.020
0.2
0.015
0.010
0.1
V/I_c
U/I_c
0.005
0.000
0.0
−0.005
−0.1
−0.010
−0.015
−0.2
−0.020
−0.2
−0.1
0.0
0.1
0.2
−0.2
Wavelength (nm)
−0.1
0.0
Wavelength (nm)
Figure 4: Synthetic Stokes profiles of Stokes I (top left), Q (top right), U (bottom left) and V
(bottom right) for the Ca i line at 526.171 nm in the synthesis with 1500 G.
8
V amplitudes for all lines from the 1500 G synthesis (see Tables 4 and 5). I identified a subset
of six lines with or without linear polarization (cf. Table 1) for testing the approach of inferring
polarimetric telescope properties. These lines were selected because of their large line depths and
polarization amplitudes, the absence of line blends (cf. Figs. 2–4, Figs. 18–19), and a continuous
wavelength coverage from 400–615 nm. Figures 2 to 4 show the synthetic profiles of the lines at 459,
526 and 615 nm analyzed in this study. Neutral lines, such as Ca i at 526.2 nm, are generally better
suited than singly-ionized lines because the latter disappear in a cool atmosphere such as present
in sunspots and pores. The Ca i line at 526 nm has small LP (ratio of V /LP ≈ 10) while the
lines at 459 and 615 nm have zero LP. The only line that has been previously used for calibration
purposes is Fe ii at 614.9 nm (Lites 1993; Cabrera Solana et al. 2005; Elmore et al. 2010).
Table 1: Transition parameters, line depth and maximal polarization amplitudes at 1500 G.
Element
Mn
Ti
Cr
Fe
Ca
Fe
4
Ion.
State
I
II
II
I
I
II
λ0
[nm]
425.766
431.490
458.820
514.174
526.171
614.923
Exc.
Pot. (eV)
2.953
1.1609
4.071
2.424
2.521
3.889
log (gf)
-0.70
-1.104
-0.64
-2.238
-0.73
-2.8
Transition
4D 0.5- 4P
4P 0.5- 4D
4D 0.5- 6F
3P 1.0- 3D
3D 1.0- 3P
4D 0.5- 4P
0.5
0.5
0.5
1.0
1.0
0.5
geff
LP?
1.333
1.333
0.333
1.000
1.000
1.333
no
no
no
low
low
no
Line
Depth [%]
81.0
92.1
81.2
84.3
79.7
45.6
Max.
|Q/I|
0
0
0
0.003
0.003
0
Max.
|U/I|
0
0
0
0.011
0.008
0
Application of Lines without LP for Polarimetric Calibration
~ after propagation through the telescope
The following expression is valid for the Stokes vector S
optics for a spectral line without LP where T is the Mueller matrix of the telescope and (I ′ , 0, 0, V ′ )T
is the light incident on the telescope:
 ′
  
T11 T12 T13 T14
I
I
Q T21 T22 T23 T24   0 
 
~= =
(1)
S
U  T31 T32 T33 T34   0 
V
T41 T42 T43 T44
V′
The observed linear and circular polarization are then given by:
Q = T21 · I ′ + T24 · V ′
(2)
U
′
= T31 · I + T34 · V
′
(3)
V
′
′
(4)
= T41 · I + T44 · V .
The cross-talk I → QU V ≡ Ti1 · I can easily be measured at continuum wavelengths outside of any
spectral line and subsequently be compensated for, which yields:
Qcorr = T24 · V ′
(5)
′
(6)
′
(7)
Ucorr = T34 · V
Vcorr = T44 · V .
With that, the ratios Qcorr /Vcorr and Ucorr /Vcorr correspond to the ratios of telescope matrix entries
T24 /T44 and T34 /T44 . With a determination of Qcorr /Vcorr and Ucorr /Vcorr from observational data
taken at a given telescope, one can thus obtain the ratio of telescope matrix entries at different times
9
Max.
|V /I|
0.216
0.156
0.074
0.138
0.127
0.134
and telescope geometries. If a model of the polarimetric properties of the corresponding telescope
is available, one can then infer the set of model parameters that best reproduce the observed
Qcorr /Vcorr and Ucorr /Vcorr ratios. The approach can also be used in the opposite direction for test
purposes such that if a telescope correction is applied using a set of parameters and there is some
residual LP, it can be deduced that the model parameters are incorrect and need to be updated.
For lines with small linear polarization, Eqs. (5)–(7) are not strictly valid, but with V /LP ≈ 10
the contribution from T22 , T23 , T32 and T33 is still very small.
5
Observations
To test the usage of spectral lines without linear polarization for the inference of telescope
parameters, V. Pillet, C. Beck, and myself ran an observing campaign at the Dunn Solar Telescope
(DST) in Sunspot, NM on March 8–31, 2016. We combined the Spectropolarimeter for Infrared
and Optical Regions (SPINOR; Socas-Navarro et al. 2006), the Facility Infrared Spectropolarimeter
(FIRS; Jaeggli et al. 2010) and the Interferometric Bidimensional Spectrometer (IBIS; Cavallini
2006; Reardon & Cavallini 2008) in order to obtain spectropolarimetric data in different wavelength
regions at the same time. A dichroic beam splitter (BS) was used to separate infrared (IR) and
visible (VIS) wavelengths. FIRS was fed with the IR light while the VIS light was split evenly
between IBIS and SPINOR by an achromatic 50-50 BS. IBIS and FIRS observed spectral lines with
a regular Zeeman pattern (617.3 nm, 630.25 nm, 656 nm, 854 nm, 1083 nm and 1565 nm) which were
not used for the current study. Both IBIS and FIRS were subsequently dropped from the setup
on March 17 and 18, respectively, and the corresponding BSs in the light feed were removed or
replaced with flat mirrors to increase the light level in SPINOR.
SPINOR was set to observe the six lines with small or without LP listed in Table 1 in two
different configurations. In the first configuration (setup 1), the lines at 426 nm, 431 nm, and
615 nm were observed from March 8-17, while the lines at 459 nm, 514 nm, 526 nm were covered
from March 17-24 (setup 2). The wavelength range always covered additional lines with a regular
Zeeman pattern as well (Figs. 5–7, Figs. 20–22). The spectral sampling was done between 3–5 pm.
The exposure time was between 22–125 ms and the integration time was between 10–30 s depending
on the setup of SPINOR and the light feed. The spatial sampling in the scanning direction was
0.′′ 75 (0.′′ 37) in setup 1 (setup 2) and the sampling along the slit was about 0.′′ 35 in all cases. The
slit width was 100µm corresponding to about 0.′′ 75 on the Sun.
In each configuration, several maps of the sunspots in NOAA 12519 and 12524 with 30′′ × ≈ 100′′
extent were acquired each day. We also took a few observations of small pores and quiet Sun at
disc center to sample regions with smaller polarization amplitudes for comparison. In total, about
30 data sets were obtained in each setup.
Given the amplitude of the interference fringes at all wavelengths shorter than 450 nm (Figs. 20
and 21) and the fact that 514 nm and 526 nm are close to each other, we only analyzed the data at
459 nm, 526 nm and 615 nm.
6
6.1
Data Analysis
Preparation of Input Data
This section addresses the methods I used to infer the parameters of the DST telescope model
from the observations and the steps required to prepare input data corresponding to Equations
(5)–(7) for all locations with significant polarization signal.
10
6.1.1
Reduction of SPINOR Data
I first reduced the data without applying the correction for the telescope polarization, but with
a correction for I → QU V cross-talk, using the standard SPINOR data pipeline10 . The I → QU V
cross-talk was determined in a continuum wavelength range close to the line of interest without
linear polarization. The values of the I → QU V cross-talk were stored for later use because
they contain the information on the first column of the telescope matrix. After this step, a twodimensional (2D) Fourier filtering in Stokes QU V was applied in order to reduce the amplitude of
interference fringes. The Fourier filter was set to remove spectral frequencies within a manually
set frequency range that were constant along the slit, i.e., only fringes with a spatial frequency of
zero were removed. The fringe pattern in all of the data also contained higher-order fringes that
were not taken out. Since the polarization signal can easily extend along 50 % of the slit length in
observations of active regions, any attempt to include higher spatial frequencies can remove genuine
polarization signal as well. The fringe amplitude, period and phase were also not constant across
the spectrum in both the spatial and spectral domain, thus only the primary fringe component
could be corrected for. The example slit spectra in Figures 5–7 show spectra after the correction
for the strongest fringe pattern.
6.1.2
Determination of Locations with Significant Polarization Signal
To determine the locations inside the field of view (FOV) that showed significant polarization
signal, the maximal polarization degree, pmax , of every profile was calculated by
p
(8)
pmax = max[p(λ)]|∆λ = max[ Q2 + U 2 + V 2 /I(λ)]|∆λ
in a small wavelength range ∆λ around a spectral line with strong polarization signal. The line
need not have been the one with low or zero LP as each spectral range also covered stronger lines
(Figures 5–7).
From the analysis of all profiles, spatial maps of the polarization degree in 459 nm, 526 nm, and
614 nm were created for all observations (see Figure 8). These maps were used to determine the
locations of significant polarization signal through a threshold value of 0.01–0.03 by rejecting all
profiles with lower polarization amplitudes. In addition, by requiring that the profiles above the
threshold belong to a spatially connected area with some minimal size, I masked out everything
but sunspots and pores. For the 459 nm data, in several cases, the fringe pattern in the polarization
signal was too strong to use the maps of pmax . Instead, a threshold in the map of the continuum
intensity was used to separate the umbra and penumbra of the sunspot from the brighter quiet Sun
and to only retain the profiles located inside the sunspot.
6.1.3
Determination of V → QU Cross-Talk
A determination of the V → QU cross-talk from the ratio of observed profiles Q/V and U/V
can become unreliable for small values of V in the presence of noise. Therefore, a linear fit of
QUobs (λ) = αQU Vobs was made for all profiles above the polarization threshold (see, e.g., Figure
9). Only the close surroundings of the spectral line of interest without LP were used to avoid
contamination with line blends.
To verify the quality of the fit, the obtained cross-talk values αQU were used to correct the
observed Q and U spectra by QU ′ = QU − αQU · V (top row of Fig. 9). The correction can be
applied to the full spectral range. In the example of Fig. 9, the change between Stokes U and U ′ is
10
http://nsosp.nso.edu/dst-pipelines
11
Figure 5: Example spectra at 458.8 nm. Top panels: individual IQU V profiles from the location
indicated with a horizontal black line in the lower panels. The red lines show the corresponding
profiles from the FTS atlas. Bottom panels: slit spectra of (clockwise, starting left bottom) IQU V
on a cut across the center of a sunspot. The line without LP is indicated by a black vertical bar.
12
Figure 6: Same as Fig. 5 for 526.2 nm.
Figure 7: Same as Fig. 5 for 614.9 nm.
13
Figure 8: Maps of one scan at 459 nm (bottom left panel), 526 nm (bottom right panel), and 614 nm
(top panel). Left to right: continuum intensity, Ic , polarization degree pmax , and mask of significant
polarization signal. The vertical bright stripes in pmax for 459 nm and 526 nm were caused by a
temporary loss of the synchronization between polarization modulation and exposures during the
scanning.
14
Figure 9: Example slit spectrum at 614 nm with the fit of V → QU . Left column, bottom to top:
Stokes I, Stokes Q, the fitted Q = αQ · V and the Stokes Q signal after correction, Q′ = Q − αQ · V .
Right column, bottom to top: Stokes V , Stokes U , the fitted U = αU · V and the Stokes U signal
after correction, U ′ = U − αU · V . The vertical dash-dotted lines indicate the location of the 614.9
nm line.
obvious, where the latter has no residual LP in the 614.9 nm line, while all other Zeeman-sensitive
lines in the wavelength range have changed from a Stokes V -like shape (a negative lobe and a
positive lobe corresponding to black and white coloring) to regular LP signals by the correction.
The fit thus correctly retrieves the V → QU cross-talk values.
The values of αQU obtained from the individual profiles were averaged along the slit over all
profiles in the mask to retrieve < αQU > (t) because all profiles along the slit were taken at the same
time. The time, t, is used as a placeholder for the telescope geometry that contains the telescope
pointing in elevation and azimuth and the position of the coudé table. The numbers used in the
calculation of the DST telescope model are actually azimuth(t), elevation(t), and table angle(t).
Figure 10 shows the values of < αQU > (t) at 614.9 nm for one of the observations. For comparison,
the values of the corresponding telescope matrix entries predicted by the “standard” DST telescope
model, i.e., using the telescope parameters determined in 2010, are overplotted in red. The modulus
of the values agrees to first order, but the agreement is not very close (cf. Sect. 6.2.2 below).
15
Figure 10: Cross-talk from V → QU averaged along the slit for the map at 614.9 nm shown in
and TT32
,
Fig. 8 (black lines). The red lines show the values of the DST telescope model for TT31
33
33
respectively, using the parameters determined in 2010.
Figure 11: Top turret mirrors of the DST telescope.
6.2
6.2.1
Fit of DST Telescope Model
DST telescope model
The polarization model of the DST has been described in detail in Skumanich et al. (1997) and
Socas-Navarro et al. (2011). The optical train consists of the entrance window to the evacuated
steel tube, the two flat turret mirrors at an angle of incidence (AOI) of 45 degrees in an alt-azimuth
16
Figure 12: Input to the fit of telescope parameters derived from all observations of 526.2 nm during
the campaign. Left column, top to bottom: observed I → QU V cross-talk (black crosses). Right
column, top to bottom: V → QU cross-talk. The red crosses show the corresponding values of the
DST telescope model using the 2010 model parameters.
17
Figure 13: Magnification of the V → QU cross-talk at 526.2 nm for observations taken between UT
20:00 - 22:00. Individual maps lasted about 30 min. Most scans show a dominant parabolic shape
of the cross-talk values (seen also in Fig. 12). Only the central position of the scan yields a value
that agrees with the 2010 telescope model.
mounting, the primary mirror at an AOI of less than 1 degree and the exit window of the vacuum
tube (see Fig. 11). Both turret mirrors are described with the same set of parameters, the ratio of
reflectivities parallel and perpendicular to the plane of reflection, X = rs /rp , and the retardance
τmirror. The primary mirror is modeled as an ideal mirror because of the small AOI. Entrance
(EN) and exit (EX) windows are modeled independently as ideal retarders with two parameters
each, the retardance τEN,EX and the orientation of the fast axis βEN,EX . To capture the orientation
of the telescope model relative to the instrument calibration unit that is located on the rotating
coudé table, an offset angle θoffset between the telescope model and coudé table is used as a free
parameter. The known geometry of the telescope at any given moment in time is expressed by the
elevation, azimuth and the relative orientation of the coudé table (table angle).
The free parameters of the DST telescope model are usually determined from measurements
with a telescope calibration unit (TCU). The TCU consists of an array of sheet polarizers that is
placed on top of the entrance window. The TCU is motorized and can be rotated in a full circle.
The evaluation of the data and the results are described in detail in Socas-Navarro et al. (2011).
In the current study, I used the parameter set that was determined using the TCU in 2010 as a
reference.
6.2.2
Input Data for Fit of Telescope Model
The input data for the fit of the telescope model parameters have two components (see Fig 12);
the values of the I → QU V cross-talk that correspond to the first column of the telescope matrix,
and the values of the V → QU cross-talk that correspond to the ratios of the matrix entries T24 /T44
and T34 /T44 . Both quantities were determined in individual profiles and then averaged along the slit
18
because each slit spectrum corresponds to one moment in time. The geometrical input parameters
(elevation, azimuth, table angle) were extracted from the file headers of each spectrum. In the
following figures, I plot all the values just as a function of time for simplicity. The data of each
wavelength range were taken within about a week, so the solar position was similar at the same
time on different days. All calculations, however, always used the actual telescope geometry at the
moment of the observations.
After plotting the corresponding curves for the data at any of the three wavelengths, I noticed
a parabolic pattern in the V → QU cross-talk data which did not match the 2010 telescope model,
but rather touched its predicted values only at one scan step (Fig. 12). The parabolic shape is very
obvious for some of the individual maps (Fig. 13). It presumably was caused by the specific setup
for the observations. The SPINOR modulator had been removed from its location in the collimated
beam upstream of all instruments and had been placed as close to the slit as possible to prevent
interference with the polarization measurements with FIRS and IBIS, and to minimize the image
motion caused by a wedge in the modulator optics. The spatial scanning of SPINOR is achieved by
moving part of the spectrograph, i.e., the slit unit, the first fold mirror and the collimator, laterally.
The spatial scanning therefore moved the slit relative to the modulator, sampling different areas
on the modulator depending on the scan position. The polarimetric calibration of the instrument,
however, was only done with the slit centered. Together, this led to a variation of the polarization
modulation across the FOV that was not removed by the calibration process. Thus, all of the data
points apart from the central slit position of each map were dropped.
6.2.3
Fit of Telescope Model Parameters
I then attempted to fit the open parameters of the DST telescope model, X and τ for the turret
mirrors, the retardance τEN,EX , and the orientation of the fast axis βEN,EX of the entrance and exit
windows to the observed I → QU V and V → QU cross-talk values considering only the central
step of each observation.
It was immediately obvious that there is a sign conflict between the observations and the model
in the I → QU V cross-talk (orange and red or blue pluses in Fig. 14) that were opposite of each
other. A mismatch of signs between the 2010 model values and the more recent SPINOR data
pipeline was already known. For the application of the telescope correction using the 2010 values,
the offset angle θof f of about 90 degrees in 2010 had to be changed to θof f + 90 ∼ 180 degrees.
There are unfortunately multiple reasons that could cause this sign conflict. A difference in the
selection of beams when merging the Stokes vectors would flip the signs of Stokes Q, U and V .
The linear polarizer of the calibration unit (CU) was taken out and installed back into the CU a
few times between 2010 and 2016. If the optical axis got switched by 90 degrees when replacing
the polarizer, that also would flip the signs in some of the Stokes parameters. V. Pillet, C. Beck,
and myself were not able to identify the source for the sign mismatch. We therefore used an ad-hoc
correction by flipping the sign of the observed I → QU V cross-talk values and used an offset angle
of 180 degrees as initial value. The V → QU values are not affected because they represent a ratio
of Stokes parameters.
A second point that became obvious instantly was that the retardances of the windows were
not constrained by the data. When trying to fit the retardances of the windows and of the turret
mirrors at the same time, the former were running off to unreasonably large values of more than
20 degrees in retardance. Depending on the initial values of the fit, the retardances of the two
windows were sometimes changing in opposite directions with sort of a zero net retardance (see
also Socas-Navarro et al. 2011, their Sect. 3).
Therefore the fit was modified to a two-step procedure. In the first step, only X, τ and θof f
19
Figure 14: Fit of telescope parameters at 458.8 nm, 526.2 nm, and 614.9 nm. Top to bottom:
observed cross-talk of V → Q (black crosses), V → U , I → Q, I → U , and I → V . The blue (red)
crosses show the corresponding values of the DST model using the 2016 (2010) model parameters.
The orange pluses in the lower three panels of the leftmost column show the original I → QU V
values before flipping their sign.
20
Table 2: Literature values, best-fit results and error estimates of telescope parameters.
X
τ
θof f
τEN
βEN
[deg] [deg] [deg] [deg]
458.8 nm
Literature
1.057
165.8
–
–
–
1.101
148.9 93.1
3.0
138.6
2010 TCU
2016 fit
1.077
153.2 178.2
0.6
22.0
± 0.033 ± 0.5 ± 0.7 ± 0.6
–
σ
526.2 nm
Literature
1.059
167.7
–
–
–
2010 TCU
1.083
155.0 93.1
2.3
138.6
2016 fit
1.108
154.3 182.0
0.6
22.0
± 0.035 ± 0.5 ± 0.7 ± 0.6
–
σ
614.9 nm
Literature
1.065
169.5
–
–
–
2010 TCU
1.080
157.8 93.1
2.1
138.6
1.074
162.7 174.2
0.4
22.0
2016 fit
σ
± 0.036 ± 0.6 ± 1.1 ± 0.8
–
were allowed to vary, while in the second step the retardance, position angle of the entrance window
and θof f were varied. The exit window was not considered due to its small size and set to a unity
matrix. The retardance of the mirrors was found to be already able to reproduce the observations
very well, so that the window’s retardance in the second step stayed close to zero. Note that in the
DST telescope model, the window retardance to some extent only gets added to that of the mirrors,
with the orientation of its optical axis as one more degree of freedom. The final fit procedure in
our case then only used X, τ , θof f , βEN and τEN .
7
7.1
Results
Fit Quality
Figure 14 shows the final result of the fit (blue pluses) together with the corresponding curves
when using the 2010 model parameters (red pluses) and the observations (black pluses) for all three
wavelengths. The input data (observations) at 458.8 nm shows the largest scatter that presumably
resulted from the residual fringe pattern (Fig. 5). The differences between the new determination
(the fit) and the 2010 values are rather small, only a few percent, however both parameter sets
provide a good fit to the input data. The match of the observations and the telescope model with
any parameter set is slightly worse in the afternoon for times later than about UT 19:00.
7.2
Ratio of Reflectivities X = rs /rp
Table 2 lists the results for all free fit parameters from the current determination and the 2010
measurements. In addition, I calculated the values of X and τ for the turret mirrors assuming
optically thick coatings (i.e. light does not penetrate the aluminum coating of the mirror and there
is no influence of the underlying glass substrate) using the literature values at each wavelength
given in the Handbook of Chemistry and Physics. For the ratio of reflectivities X, there is no clear
wavelength trend in either the 2010 or 2016 results, while the literature values show a monotonic
21
decrease with increasing wavelength (see Fig. 15). All values for X determined from measurements
at the DST are around 1.09±0.1, while the literature values are about 0.02-0.05 lower (see also
Socas-Navarro et al. 2011, their Fig. 8). The value of X is only weakly constrained by the V → QU
cross talk, where it cancels out for a single mirror,
V → Q ≡ T24 /T44 =
−2 X sin τ
= tan τ 6= f(X) .
2 X cos τ
(9)
The sensitivity test described in Sect. 7.4 below confirms this. Within the assumed error range,
the measurements of X from data acquired at the DST agree well.
7.3
Window and Mirror Retardance τ
The mirror retardance alone was found to already cover the needed value and to provide a
good fit. The second step of the two-step fit using the entrance window retardance as described
in Sect. 6.2.3 above yielded a window retardance τEN of less than 1 degree, therefore making the
window seem negligible. Note that the initial value of the orientation of the optical axis of the
window for the fit was 22 degrees. This was not significantly changed in the fit (last column
of Table 2), which would support the assumption that the window does not contribute to the
polarization properties of the optical train, or its contribution cannot be separated from that of
the mirrors given the input data.
The retardance of the turret mirrors τ increases monotonically with wavelength in all cases,
i.e., for the 2010 fit, the 2016 fit, and the literature values (see Fig. 15). The 2010 and 2016 fit
have rather similar values, while the literature values are offset from them by about 5 degrees. The
literature values do not take any information obtained at the DST into account, so their global
offset from both fit values is uncritical, whereas the fit values themselves agree well.
Figure 15 shows a plot of both X and τ for the turret mirrors for all available values for
completeness. The error bars were taken from the estimates in the next section.
7.4
Sensitivity Test and Error Estimate
To check the sensitivity of the fit and the telescope model to changes in the model parameters,
I varied X and τ of the turret mirrors by -0.1 to 0.1 and -10 to 10 respectively around the best-fit
value, while keeping the window properties at the best-fit values. Figure 16 shows the reaction of the
telescope model in the I → QU V and V → QU cross-talk values to these changes at a wavelength
of 614.9 nm. The other two wavelengths at 459 nm and 526 nm showed a similar behavior.
The V → QU cross-talk reacts only weakly to changes in X. Only for values of X that were
far outside the range of variation used here, e.g., X < 0.5 or X > 2, a significant change was seen.
This is due to the fact that the DST contains two mirrors that rotate relative to each other when
tracking the Sun, while X cancels out for a single mirror as shown by Eq. (9) above. The I → QU V
cross talk on the other hand reacts strongly to changes in X, while it is completely unaffected by
changes in τ .
I used the sensitivity test to estimate a reliability interval of ±3σ for the fit procedure by
determining the variation of X or τ that led to a change of the I → QU V or V → QU cross talk
by 0.01 on average. The scatter in the observational values, e.g., for I → V , is of about that order
of magnitude, so a variation by 1 % can be taken as the reliability level of the input data. This
yielded a range of about ±0.035 for X, ±(0.5 − 0.8) degrees for the retardances τ of the turret
mirrors and the entrance window, and ±1 degree for the offset angle (see Table 2). At the very
small retardance best-fit value of the entrance window of less than a degree, the orientation of the
window retardance was nearly unconstrained. All three spectral ranges gave very similar results.
22
Figure 15: Values of X (top panel) and τ (bottom panel) for the literature values (dashed lines),
the 2010 telescope model parameters (asterisks), and the current 2016 fit results (crosses). The
vertical grey lines show error estimates of ± 0.035 (top panel) and ± 1 degree (bottom panel).
Table 3: Residual rms V → QU cross talk.
V →Q V →U
458.8 nm
Uncorrected
0.27
0.43
0.11
0.12
2010 TCU
2016 fit
0.10
0.12
526.2 nm
Uncorrected
0.33
0.47
2010 TCU
0.04
0.06
0.03
0.05
2016 fit
614.9 nm
Uncorrected
0.17
0.25
0.08
0.11
2010 TCU
0.09
0.11
2016 fit
7.5
Residual cross talk with telescope correction
To test the fit quality, I applied the telescope correction to the observations using the parameter
sets from 2010 and 2016 prior to running the determination of the I → QU V and V → QU crosstalk
in the same way as before. With the telescope correction, all cross-talk values should be close to
zero. Figure 17 shows that this is indeed the case for the V → QU cross talk. The rms fluctuations
of the V → QU cross talk are given in Table 3 for all wavelengths. With the application of the
telescope correction, they reduce from the uncorrected case with about 20–30 % rms to 3–10 %
23
Figure 16: Sensitivity test of the telescope model at a wavelength of 614.9 nm. Top three rows:
change in I → QU V for a variation of the mirror retardance τ (left column) and the ratio of reflectivities X (right column). Observations are given by black pluses, while the red pluses correspond
to the 2010 parameter set. τ and X were varied from -10 to 10 and -0.1 to 0.1, respectively. The
resulting curves are indicated by the series of blue to purple pluses. Bottom two rows: change in
V → QU for a variation of the mirror retardance τ (left column) and the ratio of reflectivities X
(right column).
24
Figure 17: Residual V → Q (top row) and V → U (bottom row) cross talk after applying the
telescope correction. Left to right: at 458.8 nm, 526.2 nm and 614.9 nm. Black pluses: observations
without telescope correction. Red/blue pluses: observations with a telescope correction using the
2010/2016 model parameters.
rms after the correction. The new determination in 2016 yields a slightly lower rms than the 2010
parameter set, but usually only by about 1 %.
The residual cross-talk level thus stays at a value of (3-10)×10−2 regardless of which correction is
applied. The error when applied to correct solar observations is then about one order of magnitude
lower, given that the maximal polarization amplitudes in, e.g., the solar photosphere, are limited
to about 0.3 in the very umbra of sunspots by the thermal broadening.
8
Summary & Discussion
I acquired a series of spectropolarimetric observations of active regions containing sunspots
over a few days in six different spectral lines with or without negligible linear polarization signals
(cf. Table 1). From the analysis of the cross-talk between intensity and polarization, I → QU V ,
and the cross-talk between circular and linear polarization, V → Q and V → U , I was able to infer
the parameters that characterize the model of the polarization properties of the DST for a subset
of three spectral lines, and hence three wavelength regions at 459, 526 and 615 nm (cf. Table 2).
8.1
Method
Unlike most other calibration techniques in use for current ground-based solar telescopes (Skumanich et al. 1997; Beck et al. 2005; Selbing 2010; Socas-Navarro et al. 2011), this approach does
not require any additional optics other than a spectropolarimeter with medium to high spectral
resolution. The method does not rely on any assumptions for lines without intrinsic linear polarization, while for the other lines the linear polarization is assumed to be negligibly small (Lites
1993; Sanchez Almeida & Vela Villahoz 1993; Vela Villahoz et al. 1994).
The analysis procedure is automatic to a high degree. After identification of the corresponding
spectral line to be used, the thresholds in polarization degree or continuum intensity for the selection
25
of spatial positions to be included, and the initial values for the least-squares fit are the only items
which must be manually provided.
The method is based on a statistical approach. The data to be analyzed does not have to be
acquired consecutively over a few days as done in our case, but can be collected over a longer period
as long as the telescope properties did not vary significantly within that time frame. In addition,
the spatial resolution of the observations does not need to be very high because the generation of
cross-talk between polarization states only requires a significant signal in the source polarization
state. This offers the possibility to build up a data set with sufficient statistics over a time frame
of weeks to months during periods of bad seeing.
In the same way that high spatial resolution is not ultimately required, there is also no requirement for observing active regions, pores or sunspots. Polarization amplitudes of a few percent in
Stokes V are also reached by magnetic network elements (e.g., Rezaei et al. 2007). An accurate
determination of the cross-talk between polarization states then only requires a high signal-to-noise
ratio to detect a fraction of a percent-signal, i.e., signals of 10−4 − 10−3 amplitude must be above
the noise floor.
Several of the spectral lines with or without negligible linear polarization signal that can be used
for the application of the method (see Table 4 and Table 5) result from neutral atoms, unlike the
singly-ionized line at 614.9 nm. The advantage of lines pertaining to neutral elements is that their
line strength does not weaken in “cold” solar structures such as pores and sunspots, while singlyionized lines can disappear completely exactly where the magnetic field strength and polarization
signal is largest (see Fig. 7).
Although there is a dense wavelength coverage from the blue end of the visible spectrum at
400 nm up to 615 nm, there were no suitable lines found in the red end of the spectrum (> 615 nm).
8.2
Performance
As for most methods used to determine polarization properties, it is difficult to provide a good
estimate of the accuracy of the approach. I find a residual V → QU cross-talk of about 3-10 % after
application of the correction for the telescope polarization based on our 2016 best-fit values (Table
3). The residual cross-talk is comparable to that when using the 2010 telescope parameter set that
was derived with a TCU. However, as seen in Figs. 14 and 17, there already is a significant scatter
of a few percent in the input data that is presumably caused by residual interference fringes, and
thus due to the quality of the input data, not the approach itself.
The values of the telescope parameters of my current determination match those of the 2010
telescope model within the error bars, however both are offset from literature values (Fig. 15). The
match of the values determined from actual measurements at the DST is a positive sign, as the
applicability of the literature values assuming thick coatings to the DST mirrors is not ensured.
Therefore, based on my findings, I propose that this new calibration method is just as accurate as
the standard DST calibration with a TCU.
9
Conclusions & Future Work
I conclude that it is possible to derive the parameters that describe the polarization properties
of a telescope from observations of spectral lines with or without negligible linear polarization
signal. These spectral lines cover much of the blue side of the visible part of the spectrum, but no
suitable lines were found above 615 nm. Therefore, using spectral lines without linear polarization
has proven to be a suitable tool to be used for the polarimetric calibration of large solar telescopes
such as DKIST.
26
It is impossible to use a conventional TCU consisting of linear polarizers for the 4-m DKIST
telescope. Using spectral lines without intrinsic linear polarization is thus a promising approach
for its polarization calibration. At the DST, a time-dependent system with a variable geometric
configuration has to be calibrated. The corresponding problem for DKIST is much less complex as
only the first two mirrors need to be characterized which are static with fixed angles of incidence
and a fixed relative orientation.
I would also like to search for suitable lines towards the red end of the spectrum and develop
of a similar method to be used on lines with a regular Zeeman pattern, for which one will need to
make either assumptions about the magnetic field geometry or the symmetry properties of Zeeman
signals.
10
Acknowledgments
Thank you to my mentors Christian Beck and Valentin Pillet of the National Solar Observatory
who made this project possible.
The National Solar Observatory is operated by the Association of Universities for Research in
Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation.
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28
A
Appendix A: Line spectra at 426, 431 and 514 nm
1.0
1.0
0.8
0.8
0.6
0.6
I/I_c
1.2
I/I_c
1.2
0.4
0.4
0.2
0.2
0.0
0.0
−0.2
−0.2
−0.2
−0.1
0.0
0.1
0.2
−0.2
−0.1
0.04
0.04
0.02
0.02
0.00
−0.02
−0.04
−0.04
−0.1
0.0
0.1
0.2
0.1
0.2
0.00
−0.02
−0.2
0.0
Wavelength (nm)
V/I_c
V/I_c
Wavelength (nm)
0.1
0.2
−0.2
Wavelength (nm)
−0.1
0.0
Wavelength (nm)
Figure 18: Synthetic Stokes profiles of Stokes I (top panel) and V (bottom panel) for the Mn i line
at 425.766 nm (left column) and Ti ii at 431.490 nm (right column) in the synthesis with 1500 G
shown in red. The blue lines show the corresponding profiles from the FTS atlas.
Figures 18 and 19 show the profiles of the lines of Mn i at 425.766 nm (no LP), Ti ii at
431.490 nm (no LP) and Fe i at 514.174 nm (small LP) from the synthesis with a magnetic field
strength of 1500 G.
Corresponding example spectra from the observations are shown in Figs. 20 to 22. The Ti ii
line at 431.5 nm is partly blended with other Zeeman-sensitive lines. The residual fringe amplitude
of still a few percent after removal of the strongest interference fringes makes any evaluation of the
data at 425.8 nm and 431.5 nm unreliable. The observed peak Stokes V polarization amplitudes
of both lines of about 0.1 are sufficient for the inference of telescope properties as demonstrated
with the Ca i line at 526.2 nm that has a maximal Stokes V amplitude of 0.127. The Fe I line
at 514.2 nm is well isolated in the spectrum (Figs. 19 and 22) and does not suffer strongly from
interference fringes in the observations, but its wavelength range was already covered by the Ca i
line at 526 nm, so we did not evaluate this data.
29
1.2
0.020
0.015
1.0
0.010
0.8
0.005
I/I_c
Q/I_c
0.6
0.000
0.4
−0.005
0.2
−0.010
0.0
−0.015
−0.2
−0.020
−0.2
−0.1
0.0
0.1
0.2
−0.2
−0.1
Wavelength (nm)
0.0
0.1
0.2
0.1
0.2
Wavelength (nm)
0.020
0.2
0.015
0.010
0.1
V/I_c
U/I_c
0.005
0.000
0.0
−0.005
−0.1
−0.010
−0.015
−0.2
−0.020
−0.2
−0.1
0.0
0.1
0.2
−0.2
Wavelength (nm)
−0.1
0.0
Wavelength (nm)
Figure 19: Synthetic Stokes profiles of Stokes I (top left), Q (top right), U (bottom left) and V
(bottom right) for the Fe i line at 514.174 nm in the synthesis with 1500 G.
30
Figure 20: Example spectra of Mn I at 425.8 nm. Left four panels: individual IQU V profiles
from the location indicated with a horizontal black line in the right panels. The red line shows the
corresponding profile from the FTS atlas. Right four panels: slit spectra of (clockwise, starting
left bottom) IQU V on a cut across the center of a sunspot. The line without LP is indicated by a
black vertical bar.
Figure 21: Same as Fig. 20 for Ti II at 431.5 nm.
31
Figure 22: Same as Fig. 20 for Fe I at 514.2 nm.
B
Appendix B: Transition parameters and polarization amplitude
Tables 4 and 5 contain the transition parameters and polarization amplitudes in the synthesis
with 1500 G for all lines without and with small intrinsic linear polarization, respectively. All
spectral lines for which the maximal Stokes V amplitude in the 1500 G synthesis is about 10 % or
more are suitable for the inference of telescope properties as demonstrated with the Ca i line at
526.2 nm. For lines with lower polarization amplitudes, the problem reduces to the question whether
the cross-talk is large enough that the spurious signals can still be measured in the observations.
For the case of the DST, the V → QU cross-talk can be huge with values of Ti4 /T44 above 1, but for
other telescope that number can be as small as only a few percent. This implies that polarization
signals at the 10−4 − 10−3 level have to be measured.
32
Table 4: Spectral lines without linear polarization. Transition parameters,
sity and maximal Stokes V polarization amplitudes at 1500 G.
Element Ion.
λ0
Excitation
log (gf)
Transition
State
[Å]
Potential (eV)
Cr
II
4087.59
3.103
-3.22
4D 0.5- 6D 0.5
V
I
4093.5
1.183
-1.02
4P 0.5- 4D 0.5
V
I
4153.32
0.262
-2.55
6D 0.5-4D 0.5
Cu
I
4248.95
5.076
-0.976 4P 0.5- 4D 0.5
Mn
I
4257.66
2.953
-0.70
4D 0.5- 4P 0.5
S
II
4278.5
16.092
-0.11
4P 0.5- 4D 0.5
Ti
II
4314.9
1.1609
-1.104 4P 0.5- 4D 0.5
Fe
II
4385.38
2.778
-2.6
4P 0.5- 4D 0.5
Ti
II
4407.67
1.221
-2.617 2P 0.5- 4D 0.5
Ti
II
4411.93
1.224
-2.524 4P 0.5- 4D 0.5
S
II
4456.38
15.848
-0.55
4D 0.5- 4P 0.5
Cr
II
4588.20
4.071
-0.64
4D 0.5- 6F 0.5
V
I
4626.48
1.043
-1.24
4D 0.5- 4P 0.5
V
I
4757.48
2.029
-0.29
6G 1.5- 6F 0.5
Co
I
5247.93
1.785
-2.08
4P 0.5- 4D 0.5
Mn
I
5292.87
3.383
-2.90
4P 0.5- 4D 0.5
S
II
5473.62
13.584
-0.226 4P 0.5- 4D 0.5
C
II
5818.31
22.528
-1.464 4D 0.5- 4P 0.5
V
I
6008.67
1.183
-2.34
4P 0.5- 4D 0.5
V
I
6111.67
1.043
-0.715 4D 0.5- 4P 0.5
Co
I
6117.00
1.785
-2.49
4P 0.5- 4D 0.5
Fe
II
6149.23
3.889
-2.8
4D 0.5- 4P 0.5
S
II
6397.99
14.154
-0.791 4D 0.5- 4P 0.5
N
I
6646.50
11.750
-1.539 4D 0.5- 4P 0.5
C
II
6787.21
20.701
-0.377 4P 0.5- 4D 0.5
F
I
6870.22
12.751
-0.27
4P 0.5- 4D 0.5
Co
I
6872.40
2.008
-1.85
4P 0.5- 4D 0.5
Cl
I
8428.25
9.029
-0.29
4P 0.5- 4D 0.5
N
I
8703.25
10.326
-0.310 4P 0.5- 4D 0.5
P
I
10596.90
6.935
-0.24
4P 0.5- 4D 0.5
N
I
11294.26
11.750
-0.531 4D 0.5- 4P 0.5
33
minimal line-core intengeff
min. I
max. |V /I|
1.667
1.333
1.667
1.333
1.333
1.333
1.333
1.333
0.333
1.333
1.333
0.333
1.333
0.167
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
0.77028
0.92144
0.98063
0.98631
0.18989
1.00043
0.07868
0.12297
0.37181
0.32543
1.00015
0.18781
0.93216
0.93489
0.75913
0.99262
1.00030
0.99982
0.99551
0.80069
0.89543
0.54384
0.99974
1.00055
1.00038
1.00011
0.76954
0.99912
0.97682
0.94471
0.99809
0.07909
0.02739
0.00708
0.00487
0.21631
2.6e-17
0.15555
0.14503
0.07241
0.19735
1.2e-07
0.07415
0.02359
0.00456
0.08375
0.00269
3.7e-06
9.9e-13
0.00142
0.06834
0.03575
0.13369
2.4e-07
4.9e-05
9.7e-11
5.6e-08
0.08110
0.00046
0.00858
0.01941
0.00062
Table 5: Spectral lines with small linear polarization. Transition parameters and maximal Stokes
QU V /I polarization amplitudes at 1500 G.
Elem.
Ti
Fe
Fe
He
Fe
Fe
Co
Al
Fe
Cr
Fe
Cl
Sc
Cr
Fe
Cr
Fe
Cr
Ca
He
Ni
Cr
Al
P
Cr
Ti
C
Cr
Cl
Ni
Fe
Si
Fe
Sr
Cl
Cl
P
Fe
Cr
Ti
Ti
Fe
Fe
Fe
Ion.
State
I
I
I
I
I
I
I
II
I
I
II
II
II
I
I
I
I
I
I
I
I
II
II
II
I
I
I
I
II
I
I
I
II
I
II
II
II
I
I
I
I
I
I
I
λ0
[Å]
4005.95
4007.27
4017.08
4026.19
4076.22
4109.80
4132.14
4227.95
4229.51
4232.23
4296.57
4304.04
4305.71
4312.47
4367.90
4410.96
4422.57
4429.92
4435.69
4471.48
4513.0
4539.59
4589.67
4626.70
4698.94
4722.61
4738.21
4767.27
4778.91
4808.87
4813.11
4823.32
4833.19
4876.08
4922.15
4924.25
4927.20
4930.32
4936.34
4941.57
4997.09
5021.59
5099.08
5141.74
Excit.
Pot. (eV)
2.103
2.759
2.759
20.96
3.071
2.845
1.049
15.062
3.274
4.207
2.704
15.712
0.595
3.113
1.608
2.983
2.845
3.556
1.886
20.964
3.706
4.0423
15.062
12.812
3.079
1.053
7.946
3.556
17.086
3.706
3.274
4.930
2.657
1.798
15.714
15.626
12.791
3.960
3.113
2.160
0.000
4.256
3.984
2.424
log (gf)
Transition
geff
max. |Q/I|
max. |U/I|
max. |V /I|
-0.53
-1.276
-1.992
-1.453
-1.99
-0.940
-2.82
-1.709
-1.628
-0.50
-2.9
-0.68
-1.30
-1.37
-2.886
-1.22
-1.115
-0.67
-0.519
-1.036
-1.47
-2.53
-1.608
-0.32
-1.44
-1.33
-3.115
-1.02
-0.35
-1.41
-2.84
-2.33
-4.8
-0.551
-0.59
-1.54
-0.68
-1.201
-0.34
-1.01
-2.056
-0.677
-1.265
-2.238
5F 3.0- 5H 4.0
3G 3.0- 3F 2.0
3G 3.0- 3D 2.0
3P 1.0- 3D 1.0
3P 1.0- 3D 1.0
3P 1.0- 3D 1.0
2F 2.5- 4D 2.5
3D 2.0- 3F 2.0
3D 1.0- 3P 1.0
3D 2.0- 3G 3.0
4P 1.5- 4F 2.5
3D 1.0- 3P 1.0
3F 2.0- 3D 2.0
3F 3.0- 3H 4.0
3F 2.0- 5G 2.0
3H 5.0- 5F 4.0
3P 1.0- 3D 1.0
3D 1.0- 3P 1.0
3P 1.0- 3D 1.0
3P 1.0- 3D 2.0
3D 2.0- 3F 2.0
2F 2.5- 4D 2.5
3D 2.0- 3F 2.0
3D 2.0- 3F 2.0
3G 3.0- 3D 2.0
3P 1.0- 3D 1.0
3D 1.0- 3P 1.0
3D 2.0- 3F 2.0
3P 1.0- 3D 1.0
3D 2.0- 3G 3.0
3D 1.0- 5D 1.0
3P 1.0- 3D 1.0
4H 5.5- 6F 4.5
3P 1.0- 3D 1.0
3D 2.0- 3F 2.0
3F 2.0- 3D 2.0
3D 1.0- 3P 1.0
3D 1.0- 5D 1.0
3F 3.0- 3H 4.0
3D 2.0- 3F 2.0
3F 2.0- 3D 2.0
5F 3.0- 5H 4.0
3F 2.0- 3D 2.0
3P 1.0- 3D 1.0
0.375
0.500
0.333
1.000
1.000
1.000
1.114
0.917
1.000
0.333
0.500
1.000
0.917
0.375
0.500
0.400
1.000
1.000
1.000
1.000
0.917
1.114
0.917
0.917
0.333
1.000
1.000
0.917
1.000
0.333
1.000
1.000
0.455
1.000
0.917
0.917
1.000
1.000
0.375
0.917
0.917
0.375
0.917
1.000
0.00025
0.01040
0.00338
1.2e-09
0.00743
0.00022
0.00212
1.1e-09
0.00782
9.9e-05
0.00579
1.9e-12
0.00252
0.00039
0.00019
0.00061
0.00165
0.00144
0.00014
9.9e-09
0.00216
0.00121
1.2e-09
4.9e-09
0.00018
0.00271
4.5e-05
0.00069
3.1e-13
0.00069
0.00349
0.00109
0.00067
0.00026
1.4e-12
1.8e-13
1.5e-09
0.00309
0.00221
0.00047
0.00432
0.00418
0.00313
0.00305
0.00025
0.02621
0.00510
1.2e-09
0.01482
0.00433
0.00199
1.1e-09
0.01728
9.7e-05
0.01172
1.9e-12
0.00369
0.00038
0.00027
0.00060
0.01134
0.00132
0.00544
9.9e-09
0.00233
0.00115
1.2e-09
4.9e-09
0.00017
0.00243
4.5e-05
0.00066
3.08e-13
0.00066
0.00309
0.00116
0.00066
0.00026
1.4e-12
1.8e-13
1.5e-09
0.00589
0.00252
0.00046
0.00508
0.00647
0.00542
0.01106
0.02517
0.20485
0.12080
1.2e-07
0.2171
0.11059
0.06178
4.9e-08
0.21710
0.01093
0.12746
9.6e-11
0.11599
0.01733
0.08157
0.02623
0.19642
0.06407
0.10952
2.1e-07
0.08957
0.04112
4.8e-08
2.0e-07
0.01415
0.09725
0.00249
0.03075
0.04350
0.04350
0.10193
0.06830
0.02097
0.00424
5.8e-11
7.4e-12
8.1e-08
0.13024
0.08351
0.02019
0.14870
0.11272
0.12185
0.13839
34
Table 6: Spectral lines with small linear polarization cont.
Elem.
Fe
Ca
Al
Ti
C
Fe
Ni
Ni
Fe
Fe
Fe
Fe
Ion.
State
I
I
II
I
I
I
I
I
I
I
I
I
λ0
[Å]
5207.94
5261.71
5280.27
5300.01
5300.87
5341.02
5353.39
5514.79
5667.66
5747.95
6085.26
6127.91
Excit.
Pot. (eV)
3.635
2.521
15.586
1.053
8.640
1.608
1.951
3.847
2.609
4.608
2.759
4.413
log (gf)
Transition
geff
max. |Q/I|
max. |U/I|
max. |V /I|
-2.40
-0.73
-1.981
-1.47
-2.622
-1.953
-2.81
-1.99
-2.94
-1.41
-2.712
-1.399
3D 1.0- 3P 1.0
3D 1.0- 3P 1.0
3P 1.0- 3D 1.0
3P 1.0- 3D 1.0
3D 1.0- 3P 1.0
3F 2.0- 3D 2.0
3P 1.0- 3D 1.0
1F 3.0- 3P 2.0
3F 2.0- 3D 2.0
3F 3.0- 3H 4.0
3G 3.0- 3D 2.0
3F 3.0- 3H 4.0
1.000
1.000
0.917
1.000
1.000
0.917
1.000
0.500
0.917
0.375
0.333
0.375
0.00485
0.00315
1.1e-10
0.00294
3.7e-05
0.00139
0.00824
0.00048
0.00253
0.00137
0.00284
0.00209
0.00421
0.00790
1.1e-10
0.00274
3.7e-05
0.00614
0.00782
0.00047
0.00357
0.00147
0.00332
0.00245
0.11170
0.12739
6.0e-09
0.07535
0.00194
0.12045
0.15667
0.01521
0.11359
0.05643
0.09539
0.07114
35